I apologize because this post goes beyond the title of this topic: instead of extrapolating concepts from the additive operation tier to the multiplicative operation tier, I am here extrapolating them another tier higher yet, up to the power tier.
Lambdas
Lambdas are the power equivalent of deltas and qoppas.
- A delta is a familiar value from the most basic tier, the additive tier: the difference between a number and the immediately preceding number in some list.
- Earlier in this thread, Dave and I introduced the qoppa, its multiplicative equivalent: the quotient between a number and the immediately preceding number in some list.
- Next, we can consider the power equivalent, a lambda, and this is the logquotient (Here is my post where I define this: viewtopic.php?t=575. I recommend you review it for my preferred notation, which I'll use in a couple places here.) between a number and the immediately preceding number in some list.
Viewing these from another angle, we can say:
- a delta tells us what number we have to add to πβ, to reach πβββ, and
- a qoppa tells us what number we have to multiply πβ by, to reach πβββ,
- a lambda tells us what number we have to raise πβ to, to reach πβββ.
Here's a table comparing deltas, qoppas, and lambdas, using the Fibonacci sequence as an example:
entry |
value |
Ξ (delta) |
Ο (qoppa) |
Ξ (lambda) |
πβ |
0 |
|
|
|
πβ |
1 |
1 |
|
|
πβ |
1 |
0 |
1 |
|
πβ |
2 |
1 |
2 |
|
πβ |
3 |
1 |
1.5 |
1.585 |
πβ
|
5 |
2 |
1.667 |
1.465 |
πβ |
8 |
3 |
1.6 |
1.292 |
πβ |
13 |
5 |
1.625 |
1.233 |
πβ |
21 |
8 |
1.615 |
1.187 |
πβ |
34 |
13 |
1.619 |
1.158 |
Note how the first several lambdas are not defined. The first defined lambda is 1.585 because the log base πβ of πβ = logβ3 (in my preferred notation, 3 /_ 2). The next lambda is 1.465 because the log base πβ of πβ
= logβ5 (5 /_ 3), etc.
I chose the name "lambdas" following the same pattern we extended for qoppas: taking the letter 'Q' for "quotient" and finding the equivalent Greek letter 'Ο' and using its name, just as someone one day took the letter 'D' for "difference" and found the equivalent Greek letter 'Ξ', using its name. A bonus for this choice is that the capital letter lambda Ξ closely resembles the capital letter delta Ξ, so we get a nice visual correspondence between the two.
At the power tier, however, there is more than one analog to deltas and qoppas. This is becauseβββunlike with addition and multiplicationβββpowers are non-commutative. In other words, while an augend and addend can be swapped with no change in sum, and a multiplicand and multiplier can be swapped with no change in product, a base and exponent can
not be swapped with no change in power. Thus, while it doesn't matter whether we consider a delta to be the addend or the augend, nor whether we consider a qoppa to be the multiplicand or the multiplier, it
does matter whether we consider the intermediate value between numeric list entries to be the base or to be the exponent. When we consider it to be the exponent (and the previous number in the list to be the base), we are looking at a specific type of exponent called a logarithm or logquotient, and so in this context it is a lambda. When we consider it to be the base (and the previous number in the list to be the exponent), we are looking at a specific type of base called a root, and so in this context it is a rho.
Here's a table extending the previous table to include rhos as well:
entry |
value |
Ξ (delta) |
Ο (qoppa) |
Ξ (lambda) |
Ξ‘ (rho) |
πβ |
0 |
|
|
|
|
πβ |
1 |
1 |
|
|
|
πβ |
1 |
0 |
1 |
|
1 |
πβ |
2 |
1 |
2 |
|
2 |
πβ |
3 |
1 |
1.5 |
1.585 |
1.732 |
πβ
|
5 |
2 |
1.667 |
1.465 |
1.710 |
πβ |
8 |
3 |
1.6 |
1.292 |
1.516 |
πβ |
13 |
5 |
1.625 |
1.233 |
1.378 |
πβ |
21 |
8 |
1.615 |
1.187 |
1.264 |
πβ |
34 |
13 |
1.619 |
1.158 |
1.183 |
Note how the first rho is not defined, like the first qoppa, on account of πβ = 0. The first defined rho is 1 because the πβα΅Κ° root of πβ is ΒΉβ1 = 1. The next rho is 2 because the πβα΅Κ° root of πβ is ΒΉβ2 = 2. The next rho is 1.732 because the πβα΅Κ° root of πβ = Β²β3. Etcetera.
I chose the name "rhos" again following the same pattern, taking the letter 'R' for "root" and finding the equivalent Greek letter 'Ξ‘' and using its name. I do recognize that "rhos" sounds like "rows", which ain't great because table data rows will often be relevant wherever rhos come up. Another minus is that the Greek letter 'Ξ‘' looks just like the Latin letter 'P'. Well, I somehow feel like lambdas are much more natural than rhos anyway, so perhaps these are acceptable problems.
Though in private correspondence with Dave, neither of us found any particular use for these analogs.
Logmod
There is also a power-tier equivalent of modulus and reduce: logdivide by π until you reach a value less than π. This would be called
logmodulus, or
logmod for short. (It could also have been "logreduce", but "logmod" just has such a nice ring to it.)
Example:
65 logmod 2 β 1.373, because
65β―β―β―β―β―β―β―β―β―β―/_ 2 β 6.022; > 2, so repeat.
β―β―6.022 /_ 2 β 2.590; > 2, so repeat.
β―β―2.590 /_ 2 β 1.373; done.
Or in other words, 65 /_ 2 /_ 2 /_ 2 β 1.373.
As with reduce, 1 will always be the lower asymptotic bound, and π will be the upper bound (not asymptotic).
But just as there were two power-tier analogs for deltasβββlambdas and rhosβββ there's a second analog for modulus, too, namely: repeatedly taking the πα΅Κ° root until less than π. Maybe this'd be called "raduce", taking "rad" from "radication", the technical term for taking the root ("radix" means "root" in Latin)? Nah, that's too cute. How about
rootmodulus, or
rootmod for short.
Example:
65 rootmod 2 β 1.685, because
65β―β―β―β―β―β―β―β―β―β―/^ 2 β 8.062; > 2, so repeat.
β―β―6.022 /^ 2 β 2.839; > 2, so repeat.
β―β―2.590 /^ 2 β 1.685; done.
Or in other words, 65 /^ 2 /^ 2 /^ 2 β 1.685.
Here's one way to think about it, in terms of agnosticism, like how absolute and undirected value are agnostic to positivity and directedness, respectively:
- Modulus π gives a number agnostic to where it's found relatively within consecutive ranges whose upper bounds are found by repeated addition: π, π+π, π+π+π, π+π+π+π...
- Reduce π gives a number agnostic to where it's found relatively within consecutive ranges whose upper bounds are found by repeated multiplication, π, πΓπ, πΓπΓπ, πΓπΓπβ¨―π...
- Rootmodulus π gives a number agnostic to where it's found relatively within consecutive ranges whose upper bounds are found by repeated power by a constant, π, π^π, (π^π)^π, ((π^π)^π)^π...
- Logmodulus π gives a number agnostic to where it's found relatively within consecutive ranges whose upper bounds are found by repeated power (of a constant), π, π^π, π^(π^π), π^(π^(π^π))...
As you can see, the ranges for the final twoβββthe two power-tier modulus operationsβββdiffer only in how the copies of π are grouped: from the right (as is the standard order of operations), or from the left.
Here's the first several of those ranges visualized, for π = 2, for all for modulus variations:
We can see that the logmodulus ranges grow even more extremely than those of the rootmodulus. If you're familiar with the concept of function growth, we can say the ranges for rootmod exhibit power growth, while the ranges for logmod exhibit an even more extreme growth known as exponential growth. Think of it this way: when we repeatedly take the previous value and raise it to the π power (grouped-from-the-left), this is a constant exponent, like power growth; when we repeatedly take the previous value and raise π to that power (grouped-from-the-right), it's a constant base, like exponential growth.
Here's a table showing off, for each modulus function with π = 1.69, the series of values repeatedly being reduced:
modulus |
reduce |
rootmodulus |
logmodulus |
10.14 |
23.29808512 |
203856.1615 |
9708370.563 |
8.45 |
13.78584918 |
1385.516624 |
30.66061426 |
6.76 |
8.15730721 |
72.26730489 |
6.52333296 |
5.07 |
4.826809 |
12.58825688 |
3.574010811 |
3.38 |
2.8561 |
4.475764394 |
2.427328364 |
1.69 |
1.69 |
2.427328364 |
1.69 |
0 |
1 |
1.69 |
1 |
And here's that graphed:
Notice that while regular modulus reduces to between 0 and π, and reduce and logmodulus reduce to between 1 and π, rootmodulus can't effectively reduce to between π and one of these identities. The logical choices are either to reduce to between π and π^π, or between π and π/^π. I think the former, for simplicity and positivity, is preferable.
What would either of these actually get used for, though? Beats me.
I haven't thought about operator symbols for logmod or rootmod, like how % is often used for plain mod. It doesn't even look like we came up with one for reduce; only the function notation redβ().
(others)
In the original post for this topic, Dave and I found superratio, subratio, and undirected value to be the multiplicative equivalents of positive, negative, and absolute value at the additive tier. I have not found there to be any meaningful analogs to these at the power tier. As Dave pointed out to me by email, the main problem here is that powers are non-commutative, and while there's an identity for exponents, which is 1 (like the multiplicative identity), there's no identity for bases, i.e. there is no constant that you can raise to the power
\(x\) and get back
\(x\). (Solve for
\(y\) in
\(x^y = x\) and you get
\(y = \log_x{x} = \frac{\ln{x}}{\ln{x}} = 1\); solve for
\(y\) in
\(y^x = x\) and you just get
\(\sqrt[x]{x}\) which has no fixed value, e.g. square root of 2 does not equal cube root of 3.) I'll leave it at this for now.
- As a final note, if readers are interested in a generalization of the relationship across the additive, multiplicative, and power tier, including the tier below addition which is called the successive tier, or the tiers above power which begin with the tetrative tier, I suggest you check out this resource: https://en.wikipedia.org/wiki/Hyperoper ... ost_common)